The use of wavelet and inverse wavelet transforms is well established in MRA (multiresolution analysis) digital data processing. A special feature of this approach is that it allows narrow windowing of short duration high frequency data while also enabling wide windowing of long duration low frequency data, as generally described in Chui, C. K., "An Introduction to Wavelets", Academic Press, Boston, Mass., 1992, which is hereby incorporated by reference's as background information.
In fact, MRA wavelet and inverse wavelet transforms are particularly useful in compression and/or decompression of 2-D data that comprises a 2-D array of data samples (or elements) having vertical and horizontal dimensions, such as 2-D image data that comprises a 2-D array of data samples representing the pixels of an image. Specifically, wavelets allow short duration high spatial frequency data sample values to be narrowly windowed and longer duration low spatial frequency data sample values to be widely windowed. Compression and/or decompression of 2-D data using wavelet and inverse wavelet transforms is generally performed in the manner described in U.S. Pat. No. 5,014,134, Lawton, W. et al., "Image Compression Method and Apparatus", issued May 7, 1991, which is hereby incorporated by reference as background information.
More specifically, in compressing original 2-D data, the original data is first decomposed over one or more resolution levels using an MRA wavelet transform. For each resolution level at which a decomposition is made, input 2-D data is decomposed with the wavelet transform into LL, LH, HL, and HH component 2-D data. The date samples of the LL, LH, HL, and HH component data respectively represent the data samples of the input data in the LL spatial frequency sub-band (i.e., having data sample values with low spatial frequency in the vertical and horizontal dimensions), the LH sub-band spatial frequency (i.e., having data sample values with low spatial frequency in the vertical dimension and high spatial frequency in the horizontal dimension), the XL sub-band spatial frequency (i.e., having data sample values with high spatial frequency in the vertical dimension and low spatial frequency in the horizontal dimension), the HI-H sub-band spatial frequency (i.e., having data sample values with high spatial frequency in the vertical and horizontal dimensions). For the first resolution level at which a decomposition is made, the original data is the input data. And, at each subsequent resolution level, the LL component data from the previous decomposition is the input data. All of the LH, HL, and HH component data from each of the decompositions and the LL component data from the last decomposition form the complete decomposed data of the original data.
The complete decomposed 2-D data is then quantized to provide quantized 2-D data. This is done by quantizing the data samples of the decomposed original data so that they only have allowable integer values. Once this is done, the quantized data is then encoded so as to compress it and provide encoded (i.e., compressed) 2-D data. This is accomplished by encoding the data samples of the quantized data based on their integer quantized values. The encoding technique used is preferably a lossless encoding technique that is highly compact, such as the those disclosed in copending U.S. patent application Nos. 08/758,589 and 08/758,590 (now U.S. Pat. No. 5,748,116), entitled "SYSTEM AND METHOD FOR TREE ORDERED CODING OF SPARSE DATA SETS" and "SYSTEM AND METHOD FOR NESTED SPLIT CODING OF SPARSE DATA SETS", filed on Nov. 27, 1996, which are hereby incorporated by reference as background information. Since the quantized data samples can only have a limited number of integer values, the amount of encoding needed to represent the data samples is reduced which therefore increases the compression ratio of the data.
Conversely, decompressing encoded 2-D data of the type just described is done in reverse order to that just described. On other words, such data is first decoded to obtain decoded 2-D data. Then, the data samples of the decoded data are dequantized to provide dequantized 2-D data. The dequantized data is then reconstructed over the same resolution levels that were used in decomposing the original data. The resulting reconstructed 2-D data is produced from the LL, LH, HL, and HH component data of the dequantized data using an inverse MRA wavelet transform that corresponds to the wavelet transform used in decomposing the original data.
In decomposing and reconstructing data, most data processing systems and methods generally use quadrature mirror filters (QMF), as described in Mallat, "A Theory for Multiresolution Data Decomposition: The Wavelet Representation", U. Penn. Report No. MS-CIS-87-22, 1987, for each corresponding resolution level at which this is done. Each QMF performs a decomposition or reconstruction at a corresponding resolution level with a set of separate low pass and high pass convolution filters.
However, since the low pass and high pass convolution filters of QMFs are separate, they do not share data with each other in computing their outputs. Moreover, these low pass and high pass convolution filters use multipliers to produce the LL, LH, HL, and HH component data described earlier. And, in QMFs used in decomposing original data, down sampling of data samples occurs only after the low pass and high pass convolution filters have processed the data samples of the data. Thus, these QMFs waste time processing data samples which are later discarded. Similarly, in QMFs used in reconstructing decomposed data, up sampling of data samples occurs before the low pass and high pass convolution filters process the data samples. Since the up sampled data samples have values of zero, these QMFs waste time processing these up sampled data samples.